Inverse Trigonometric

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Answer Key & Explanations

Inverse Trigonometric · Grade 11 · Worksheet 2

1. 2arctan(1) + arcsin(1/2) = ? Answer: 5π/6 Solution: Evaluate arctan(1). The angle whose tangent is 1 is π/4. So, arctan(1) = π/4.
   Full step-by-step solution

   Step 1: Evaluate arctan(1). The angle whose tangent is 1 is π/4. So, arctan(1) = π/4. Step 2: Multiply this result by 2: 2 * (π/4) = π/2. Step 3: Evaluate arcsin(1/2). The angle whose sine is 1/2 is π/6. So, arcsin(1/2) = π/6. Step 4: Add the two results: π/2 + π/6. Step 5: Find a common denominator to add the fractions: π/2 = 3π/6, so 3π/6 + π/6 = 4π/6. Step 6: Simplify the fraction: 4π/6 = 2π/3. The final answer is 2π/3.

2. sin⁻¹(11/61) + sin⁻¹(60/61) = ? Answer: π/2 Solution: Let α = sin⁻¹(11/61) and β = sin⁻¹(60/61). Then sin(α) = 11/61 and sin(β) = 60/61. For α, construct a right triangle with opposite side 11 and hypotenuse 61.
   Full step-by-step solution

   Step 1: Let α = sin⁻¹(11/61) and β = sin⁻¹(60/61). Then sin(α) = 11/61 and sin(β) = 60/61. Step 2: For α, construct a right triangle with opposite side 11 and hypotenuse 61. The adjacent side is sqrt(61² - 11²) = sqrt(3721 - 121) = sqrt(3600) = 60. So cos(α) = 60/61. Step 3: For β, construct a right triangle with opposite side 60 and hypotenuse 61. The adjacent side is sqrt(61² - 60²) = sqrt(3721 - 3600) = sqrt(121) = 11. So cos(β) = 11/61. Step 4: Notice that sin(α) = 11/61 = cos(β) and sin(β) = 60/61 = cos(α). This means α and β are complementary angles: α + β = π/2. Step 5: Therefore, sin⁻¹(11/61) + sin⁻¹(60/61) = π/2. The answer is π/2.

3. sin(2arctan(1/3)) = ? Answer: 3/5 Solution: Let θ = arctan(1/3). This means tan(θ) = 1/3. We need to find sin(2θ).
   Full step-by-step solution

   Let θ = arctan(1/3). This means tan(θ) = 1/3. We need to find sin(2θ). Using the double-angle identity: sin(2θ) = 2sin(θ)cos(θ). From tan(θ) = 1/3, we can construct a right triangle where the opposite side is 1 and the adjacent side is 3. The hypotenuse is sqrt(1^2 + 3^2) = sqrt(1 + 9) = sqrt(10). Therefore, sin(θ) = opposite/hypotenuse = 1/sqrt(10). And cos(θ) = adjacent/hypotenuse = 3/sqrt(10). Now substitute into the identity: sin(2θ) = 2 * (1/sqrt(10)) * (3/sqrt(10)) = 2 * 3/10 = 6/10 = 3/5. The answer is 3/5.

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). A line is drawn from the origin to the point (4,3), forming an angle θ with the positive x-axis. What is the value of θ in radians, expressed as an inverse trigonometric function? Answer: arctan(3/4) Solution: - A = (0, 0) - B = (4, 0) - C = (4, 3) This is a right triangle with the right angle at B = (4, 0) because AB is horizontal and BC is vertical.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the triangle and coordinates** The vertices are: - A = (0, 0) - B = (4, 0) - C = (4, 3) This is a right triangle with the right angle at B = (4, 0) because AB is horizontal and BC is vertical. --- **Step 2: Identify the line from origin to (4, 3)** The line from (0, 0) to (4, 3) is the hypotenuse AC. --- **Step 3: Determine the angle θ** Angle θ is the angle at the origin (0, 0) between the positive x-axis and the line to (4, 3). --- **Step 4: Use tangent ratio** In a right triangle, tan(θ) = opposite / adjacent. From the origin: - The horizontal leg goes from (0, 0) to (4, 0), length = 4 (adjacent to θ). - The vertical leg goes from (4, 0) to (4, 3), but the vertical side relative to the origin is the y-coordinate of (4, 3), which is 3 (opposite to θ). So tan(θ) = 3 / 4. --- **Step 5: Solve for θ** θ = arctan(3 / 4) Since (4, 3) is in the first quadrant, arctan gives the correct angle between 0 and π/2. --- **Final Answer:** arctan(3/4)

5. arcsin(1/2) = ? Answer: π/6 Solution: arcsin(1/2) means "find the angle whose sine is 1/2" The arcsin function returns angles between -pi/2 and pi/2 (or -90° to 90°) On the unit circle, sin(theta) = y-coordinate We need an angle where the y-coordinate is 1/2 sin(30°) = 1/2 sin(pi/6) = 1/2 pi/6 radians = 30°, which is between -pi/2…
   Full step-by-step solution

   Step-by-step solution: Step 1: Understand what arcsin means arcsin(1/2) means "find the angle whose sine is 1/2" Step 2: Recall the range of arcsin The arcsin function returns angles between -pi/2 and pi/2 (or -90° to 90°) Step 3: Think about the unit circle On the unit circle, sin(theta) = y-coordinate We need an angle where the y-coordinate is 1/2 Step 4: Recall common sine values From special triangles, we know: sin(30°) = 1/2 sin(pi/6) = 1/2 Step 5: Verify this is in the correct range pi/6 radians = 30°, which is between -pi/2 and pi/2 This is in the first quadrant, so it's within the valid range for arcsin Step 6: Check if there are other angles with sin = 1/2 While 150° (5pi/6) also has sin = 1/2, it's not in the range of arcsin The arcsin function only returns the principal value between -pi/2 and pi/2 Step 7: Conclusion Therefore, arcsin(1/2) = pi/6 Final answer: pi/6

6. A right triangle is drawn on a coordinate plane with vertices at (0,0), (3,0), and (3,4). A line is drawn from the origin to the point (3,4), creating angle θ with the positive x-axis. What is the measure of angle θ in degrees? Answer: 53.13 Solution: We have a right triangle with vertices at (0,0), (3,0), and (3,4). The line from the origin (0,0) to (3,4) makes an angle θ with the positive x-axis. Identify the sides relative to angle θ.
   Full step-by-step solution

   We have a right triangle with vertices at (0,0), (3,0), and (3,4). The line from the origin (0,0) to (3,4) makes an angle θ with the positive x-axis. Step 1: Identify the sides relative to angle θ. - The point (3,4) means the horizontal distance from the y-axis is x = 3, and the vertical distance from the x-axis is y = 4. - So, relative to angle θ at the origin: - Adjacent side (along x-axis) = 3 - Opposite side (vertical) = 4 - Hypotenuse = distance from (0,0) to (3,4) Step 2: Calculate the hypotenuse. Hypotenuse = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5. Step 3: Choose a trigonometric function to find θ. We can use tangent because tan(θ) = opposite / adjacent = 4/3. So, tan(θ) = 4/3. Step 4: Find θ using the inverse tangent (arctan). θ = arctan(4/3). Step 5: Compute arctan(4/3). Using a calculator: 4/3 = 1.333333... arctan(1.333333) ≈ 53.13010235 degrees. Step 6: Round to two decimal places. 53.13010235 → 53.13 degrees. Final answer: 53.13

7. Aroha is a surveyor mapping a steep hillside for a new walking trail. She measures the vertical drop of a section as 21 meters and the horizontal distance from the top to the bottom as 77 meters. To calculate the angle of depression of this slope, she uses inverse trigonometric functions. What is the angle of depression, in degrees, to the nearest tenth of a degree? Answer: 15.3 Solution: Identify the right triangle. The vertical drop of 21 meters is the opposite side, and the horizontal distance of 77 meters is the adjacent side relative to the angle of depression.
   Full step-by-step solution

   Step 1: Identify the right triangle. The vertical drop of 21 meters is the opposite side, and the horizontal distance of 77 meters is the adjacent side relative to the angle of depression. Step 2: Use the tangent function: tan(θ) = opposite/adjacent = 21/77 Step 3: Simplify the fraction: 21/77 = 3/11 ≈ 0.272727... Step 4: Apply the inverse tangent function: θ = arctan(21/77) = arctan(3/11) Step 5: Calculate using a calculator: arctan(3/11) ≈ 15.255 degrees Step 6: Round to the nearest tenth: 15.3 degrees The angle of depression is 15.3 degrees.

8. Liam is an aerospace engineer designing a landing approach path for a small aircraft. The plane must descend from an altitude of 3500 feet to a runway that is 7000 feet away horizontally. The descent angle θ satisfies the equation 3 tan(θ) = 1.5. What angle θ, in degrees, should Liam use for the approach path? Use an inverse trigonometric function and express your answer to the nearest tenth of a degree. Answer: 26.6 Solution: Start with the equation 3 tan(θ) = 1.5. Divide both sides by 3 to isolate tan(θ): tan(θ) = 1.5 / 3 = 0.5. Use a calculator set to degree mode: tan⁻¹(0.5) ≈ 26.56505118 degrees.
   Full step-by-step solution

   Step 1: Start with the equation 3 tan(θ) = 1.5. Step 2: Divide both sides by 3 to isolate tan(θ): tan(θ) = 1.5 / 3 = 0.5. Step 3: Apply the inverse tangent function to both sides: θ = tan⁻¹(0.5). Step 4: Use a calculator set to degree mode: tan⁻¹(0.5) ≈ 26.56505118 degrees. Step 5: Round to the nearest tenth: 26.6 degrees. The angle θ for the descent path is 26.6 degrees.